arXiv research

The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

Develops theory of relatively geometric actions on CAT(0) cube complexes.

Study equidistribution for flows on geometrically finite convergence group actions.

Lecture notes on group actions on injective spaces and Helly graphs.

Characterizes geometric actions on graphs with flexible stabilizers.

Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…

Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.

Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.

We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows tha…

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear …

Proves small cancellation free products have geometric actions on CAT(0) cube complexes.

Study of actions on curved manifolds with boundary results in new geometric invariant.

We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduct…

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

New insights into the geometry of flows on 3-manifolds.

New geometric interpretation of a group class using circle action and rotation numbers.

Study group actions on hyperbolic spaces to find algebraic and geometric properties.

Study geometrically measures to decide if modular companions are conformally equivalent.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

The theme of this survey is that subgroups of the mapping class group of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of the Teichmuller space of S, just as discrete subgroups of the isometries of hyperbolic space can be studied via their…

Study irregular behavior of ball averages for non-amenable group actions on foliations.

Groups on CAT(0) cube complexes grow exponentially uniformly.

We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.

Affine maps reveal higher rank structures in certain spaces.

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, an…

The paper explores proper actions and their relation to representation theory, with new quantitative methods.

Study proves rigidity of marked length spectra in contracting group actions.

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.

This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spa…

Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form…

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups wi…

For actions with a dense orbit of a connected noncompact simple Lie group $G$, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a $G$-action to be equivalent to one on a space of the form $(G\times K\backslash H)/Γ$, it is necessary and suff…

Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmuller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones in semi-simple Lie …

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a large…

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entrop…

Geometric quantization for specific symplectic structures proved.

Simplified Milnor-Schwarz lemma for geometric group theory.

Extending our reduction construction in \cite{Hu} to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) $T$-dua…

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subse…

Deformation spaces Hom($π$,G)/G of representations of the fundamental group $π$ of a surface $Σ$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_Σ$, preserving a Poisson structure. When $G$ is compact, the actions are ergodic. In contrast if $G$ is noncompact semisimple, the associated deformat…

The paper shows how to find inaccessible hyperbolic actions in manifold groups.

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group …

We show a geometric rigidity of isometric actions of non compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.

For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and analytic methods, essentially avoiding any deep algebraic result, it applies also t…

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…